Table Of ContentVariational Methods for Free Surface Interfaces
Variational Methods
for Free Surface Interfaces
Proceedings of a Conference Held at Vallombrosa Center,
Menlo Park, California, September 7-12, 1985
Edited by
Paul Concus and Robert Finn
Organizing Committee
R. Brown, Massachusetts Institute of Technology
P. Concus, University of California, Berkeley
R. Finn (Chairman), Stanford University
S. Hildebrandt, University of Bonn
M. Miranda, University of Trento
With 44 Figures
Springer- Verlag
New Yark Berlin Heidelberg
London Paris Tokyo
Paul Concus Robert Finn
Lawrence Berkeley Laboratory Department of Mathematics
and Department of Mathematics Stanford University
University of California Stanford, California 94305
Berkeley, California 94720 U.S.A.
U.S.A.
AMS Classification: 49FIO, 35-06, 53AIO
Library of Congress Cataloging-in-Publication Data
Variational methods for free sUlface interfaces.
Bibliography: p.
I. Surfaces (Technology)-Cong~esses. 2. Surfaces
Congresses. 3. Surface chemistry-':"'Congresses.
I. Concus, Paul. II. Finn, Robert.
T A407. V27 1986 620.1' 129 86-27899
© 1987 by Springer-Verlag New York Inc.
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ISBN-13: 978-1-4612-9101-5 e-ISBN-13: 978-1-4612-4656-5
DOl: 10.1007/978-1-4612-4656-5
Preface
Vallombrosa Center was host during the week September 7-12, 1985 to about
40 mathematicians, physical scientists, and engineers, who share a common
interest in free surface phenomena. This volume includes a selection of
contributions by participants and also a few papers by interested scientists who
were unable to attend in person. Although a proceedings volume cannot recapture
entirely the stimulus of personal interaction that ultimately is the best justification
for such a gathering, we do offer what we hope is a representative sampling of
the contributions, indicating something of the varied and interrelated ways with
which these classical but largely unsettled questions are currently being attacked.
For the participants, and also for other specialists, the 23 papers that follow
should help to establish and to maintain the new ideas and insights that were
presented, as active working tools. Much of the material will certainly be of
interest also for a broader audience, as it impinges and overlaps with varying
directions of scientific development.
On behalf of the organizing committee, we thank the speakers for excellent,
well-prepared lectures. Additionally, the many lively informal discussions did
much to contribute to the success of the conference.
The participants benefited greatly from the warm and pleasant ambience
provided by the Vallombrosa Center and its friendly and helpful staff, to whom
we wish to offer our special thanks. The conference was made possible in part
by support from the Air Force Office of Scientific Research, the Department of
Energy, the National Science Foundation, and the Office of Naval Research.
The National Science Foundation served as coordinating agency.
Paul Concus
Robert Finn
Contents
Preface..................................................................... v
List of Contributors ........................................................ ix
Optimal Crystal Shapes
JEAN E. TAYLOR and F.J. ALMGREN, JR .................................. .
Immersed Tori of Constant Mean Curvature in R3
HENRY C. WENTE.......................................................... 13
The Construction of Families of Embedded Minimal Surfaces
DAVID A. HOFFMAN ....................................................... 27
Boundary Behavior of Nonparametric Minimal Surfaces-Some Theorems
and Conjectures
KIRK E. LANCASTER ....................................................... 37
On Two Isoperimetric Problems with Free Boundary Conditions
S. HILDEBRANDT........................................................... 43
Free Boundary Problems for Surfaces of Constant Mean Curvature
MICHAEL STRUWE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53
On the Existence of Embedded Minimal Surfaces of Higher Genus with Free
Boundaries in Riemannian Manifolds
JURGEN JOST ................................................................ 65
Free Boundaries in Geometric Measure Theory and Applications
MICHAEL GRUTER . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 77
VIII Contents
A Mathematical Description of Equilibrium Surfaces
MARIO MIRANDA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 85
Interfaces of Prescribed Mean Curvature
1. T AMANINI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 91
On the Uniqueness of Capillary Surfaces
LUEN-FAI TAM ............................................................. 99
The Behavior of a Capillary Surface for Small Bond Number
DA VID SIEGEL .............................................................. 109
Convexity Properties of Solutions to Elliptic P.D.E.'s
NICHOLAS J. KOREVAAR ................................................... 115
Boundary Behavior of Capillary Surfaces via the Maximum Principle
GARY M. LIEBERMAN ...................................................... 123
Convex Functions Methods in the Dirichlet Problem for Euler-Lagrange
Equations
ILYA J. BAKELMAN ......................................................... 127
Stability of a Drop Trapped Between Two Parallel Planes: Preliminary Report
THOMAS 1. VOGEL ......................................................... 139
The Limit of Stability of Axisymmetric Rotating Drops
FREDERIC BRULOIS ......................................................... 145
Numerical Methods for Propagating Fronts
JAMES A. SETHIAN ......................................................... 155
A Dynamic Free Surface Deformation Driven by Anisotropic Interfacial
Forces
DANIEL ZINEMANAS and A VINOAM NIR .................................... 165
Stationary Flows in Visc0l!s Fluid Bodies
JOSEf" BEMELMANS ......................................................... 173
Large Time Behavior for the Solution of the Non-Steady Dam Problem
DIETMAR KRC)NER .......................................................... 179
New Results Concerning the Singular Solutions of the Capillarity Equation
MARIE-FRANC;:OlSE BIDAUT-VERON ......................................... 191
Continuous and Discontinuous Disappearance of Capillary Surfaces
PAUL CONCUS and ROBERT FINN ........................................... 197
List of Contributors
F.1. ALMGREN, JR., Mathematics Department, Princeton University, Princeton,
New Jersey 08903, U.S.A.
ILYA J. BAKELMAN, Department of Mathematics, Texas A&M University,
College Station, Texas 77843, U.S.A.
JOSEr BEMELMANS, Fachbereich Mathematik, Universitat des Saar/andes, 6600
Saarbriicken, Federal Republic of Germany
MARIE-FRAN<;:OISE BIDAUT-VERON, Department of Mathematics, University of
Tours, 37200 Tours. France
FREDERIC BRULOIS, California State University, Dominguez Hills, Carson,
California 90747, U.S.A.
PAUL CONCUS. Lawrence Berkeley Laboratory and Department of Mathematics,
University of California, Berkeley, California 94720, U.S.A.
ROBERT FINN, Department of Mathematics, Stanford University, Stanford,
California 94305, U.S.A.
MICHAEL GRUTER. University of Bonn, Mathematics Institute, 5300 Bonn,
Federal Republic of Germany
S. HILDEBRANDT, University of Bonn, Mathematics Institute, 5300 Bonn, Federal
Republic of Germany
DA VID A. HOFFMAN, Department of Mathematics, University of Massachusetts,
Amherst, Massachusetts 01003, U. S. A.
JURGEN JOST, University of Bochum, Mathematics Institute, 4630 Bochum
Querenburg, Federal Republic of Germany
x List of Contributors
NICHOLAS J. KOREVAAR, Department of Mathematics, University of Kentucky,
Lexington, Kentucky 40506, U.S.A.
DIETMAR KRONER. University of Bonn, Mathematics Institute, 5300 Bonn,
Federal Republic of Germany
KIRK E. LANCASTER, Department of Mathematics and Statistics, Wichita State
University, Wichita, Kansas 67208, U.S.A.
GARY M. LIEBERMAN, Department of Mathematics, Iowa State University, Ames,
Iowa 50011, U.S.A.
MARIO MIRANDA, Institute of Mathematics, University of Trento, 38 100 Trento,
Italy
AVINOAM NIR, Department of Chemical Engineering, Technion, Haifa 32000,
Israel
JAMES A. SETHIAN, Lawrence Berkeley Laboratory and Department of
Mathematics, University of California, Berkeley, California 94720, U.S.A.
DAVID SIEGEL, Department of Applied Mathematics, University of Waterloo,
Waterloo, Ontario N2L 3GI Canada
MICHAEL STRUWE, ETH-Zentrum, Mathematics Institute, CH-8092 Zurich,
Switzerland
LUEN-FAI TAM, Department of Mathematics, University of Illinois, Chicago,
Illinois 60680, U.S.A.
I. TAMANINI, University of Trento, Department of Mathematics, 38050 Trento,
Italy .
JEAN E. TAYLOR, Mathematics Department, Rutgers University, New Brunswick,
New Jersey 08903, U.S.A.
THOMAS I. VOGEL, Department of Mathematics, Texas A&M University, College
Station, Texas 77843, U.S.A.
HENRY C. WENTE, Department of Mathematics, University of Toledo, Toledo,
Ohio 43606, U.S.A.
DANIEL ZINEMANAS, Department of Chemical Engineering, Technion, Haifa
32000, Israel
Optimal Crystal Shapes
Jean E. Taylor and F.J. Almgren, Jr.
1. Introduction
sn n
Associated with any Borel function <I> defined on the unit sphere in R +1
with values in R u { oo} (and, say, bounded from below) and any n-dimensional
oriented rectifiable surface S in Rn+1 is the integral
r
<I>(S) = <I>(vs(x)) dHnx;
JXES
here vs( . ) denotes the unit normal vectorfield orienting S, and Hn is Hausdorff
n-dimensional surface measure. If, for example, S is composed of polygonal
Li
pieces Si with oriented unit normals Vi' then <I>(S) = <I>(v;) area(SJ Perhaps
the most important integrands <1>: S2 ~ R arise as the surface free energy
density functions for interfaces S between an ordered material A (hereafter
called a crystal) and another phase or a crystal of another orientation. In this
case vs(p) is the unit exterior normal to A at PES and <I>(S) gives the surface
free energy of S. Other interesting <I>'s need not be continuous or even bounded.
See the sailboat example of [Tl], in which <I>(v) is the time required to sail unit
distance in direction v rotated by 90°.
In this paper we survey what is known about the geometry of a single crystal
A in equilibrium. In the special case in which A is a sessile or pendant crystal
in a gravitational field and <I> is convex and invariant under all rotations about
the vertical axis, we show (for the first time) that rotational symmetrization
of A about the same axis does not increase total free energy.
We would like to acknowledge the partial support of both authors by NSF
grants.
2 Jean E. Taylor and F.J. Almgren, Jr.
2. Examples of Integrands <I>
A few examples of <1>'s are given, in order to illustrate some of the possibilities
and to provide examples for the results to follow.
EXAMPLE O. <1>o(v) = 1 for every v in sn. Then <1>(S) is the area of S.
EXAMPLE 1. <1>l(V) = IV11 + IV21 + IV31 for v = (v1, V2, v3) in S2.
EXAMPLE 2. <1>2(V) = max{lv11, IV21, Iv31} for v = (v1, V2, v3) in S2.
EXAMPLE 3. <1>3(V) = (1 - V3)1/2 + Clv31f or v = (Vi> V2, v3) in S2.
EXAMPLE 4. <1>4(V) = IV11 + IV21 + IV31 for all v = (Vi> V2, v3) in S2 except
(± 1/)3, 1/)3, 1/)3); for these v, <1>(v) = (5/6))3.
EXAMPLE 5. Let W be a compact convex body in Rn+1, and let <1>w be the
support function of W, restricted to the unit sphere. If the boundary of W is
twice differentiable and has positive upper and lower bounds on its curvatures,
the corresponding <1>w is called an elliptic integrand.
3. Free Single Crystal Problem
Given <1>, what is the shape of an open region A of volume 1 which minimizes
<1>(oA) among all regions with rectifiable boundaries having volume I? There
is a complete solution to this problem. The Wulff shape for <1> is defined to be
W<I> = {XEW+l: X' v ~ <1>(v) for all v in sn}.
Provided the interior of W<I> is nonempty, the solution to this problem (unique
up to translations) is the interior of W<I>, scaled so that its volume is 1. If the
interior of W<I> is empty, there is no solution. See [T4] for a short clean proof
of minimality, and see [Tl] for references to other proofs.
The Wulff shapes for the examples above are as follows:
W<I>o is the unit ball {x: Ixi ~ I} (which indeed is the shape of the region of
least surface area compared to any other shape with the same volume).
W<I>l is the cube {x = (X1,X2,X3): IXil ~ 1 for i = 1,2,11:.
W<I>2 is the octahedron {x: X' (± 1/)3, ± 1/)3, ± 1/-./3) ~ I}.
W<I>3 is the right circular cylinder centered at the origin with axis in the X3
direction, having radius 1 and height 2C.
W<I>4 is a cube with two of its eight corners truncated by triangular plane
segments.
W<I>w is the W used to define <1>w in Example 5.
One can extend <1> to a function on all of W+1 by defining <1>(rv) = r<1>(v) for
any nonnegative r. Since any W<I> is automatically convex and compact, the
integrands of Example 5 in fact consist of all integrands which are convex
(when so extended) and for which the free single crystal problem has a compact
